Effects of a New Triple-alpha Reaction on X-ray Bursts of a Helium Accreting Neutron Star

0.17 MB·English

by  Y. Matsuo

1
Prog. Theor. Phys. Vol. ***, No. *, May 2011, Letters
Effects of a New Triple-α Reaction on X-ray Bursts of a Helium
Accreting Neutron Star
Yasuhide Matsuo1, ∗), Hideyuki Tsujimoto1, Tsuneo Noda1, Motoaki
Saruwatari1, Masaomi Ono1, Masa-aki Hashimoto1,∗∗) and Masayuki Y.
Fujimoto2
1Department of Physics, Kyushu University, Fukuoka 812-8581, Japan
2Department of Physics, Hokkaido University,
2
Sapporo 060-0810, Japan
1
0
2 The effects of a new triple-α reaction rate (OKK rate) on the helium flash of a helium
n accreting neutron star in a binary system have been investigated. Since the ignition points
a determine the properties of a thermonuclear flash of type I X-ray bursts, we examine the
J cases of different accretion rates, dM/dt (M˙), of helium from 3×10−10M⊙ yr−1 to 3×
9 10−8M⊙ yr−1,which couldcovertheobservedaccretion rates. Wefindthatforthecases of
lowaccretionrates,nuclearburningsareignitedattheheliumlayersofratherlowdensities.

As a consequence, helium deflagration would be triggered for all cases of lower accretion
]
E rate than M˙ ≃ 3×10−8M⊙ yr−1. We find that OKK rate is consistent with the available
H observationsoftheX-rayburstsontheheliumaccretingneutronstar. Weadvocatethatthe
OKK rate is better than the previous rate for the astrophysical phenomena of X-ray burst
.
h dueto helium accretion.
p
SubjectIndex: 242,421
-
o
1. Introduction
r
st A new challenge has been given for the triple-α (3α) reaction rate, which has
a been calculated by Ogata et al.4) and found to be very large compared with the
[
previous rate used so far.5),6),7),8) As a consequence, the new rate results in the he-

4 lium (4He) ignition in the lower density/temperature on thestellar evolution of low-,
v
intermediate-, and high-mass stars,9),10) accreting white dwarfs,11),12) and accreting
4
8 neutron stars.13),14),15) Therefore, it is urgent to clarify quantitatively as possible
4
as how the new rate affects the above astrophysical phenomena, because the rate
5
plays the most fundamental role among the nuclear burning in heavenly bodies and
.
5
could do some role in the early universe, where any terrestrial experiments for the
0
1 3α reaction are very difficult. In the present paper, we investigate the effects of a
1 newly calculated 3α reaction rate (OKK rate)4) on the helium flashes that occur at
:
v the bottom layers inside the accreting envelope of a neutron star. We can use the
Xi ignition curves to findroughly whenthe nuclear ignition occurs. We note that X-ray
bursts which have been mainly studied so far are limited to the combined burning
r
a of hydrogen and helium1) and observational features such as light curves and burst
energy have been qualitatively explained well. Fujimoto et al.19) have succeeded in
simulating the X-ray bursts by solving the whole structure equations and clarified
firstly the importance of the heat flow from the bottom of the accumulated layer.
The application has been done for several X-ray burst observations of combined H
∗) E-mail: [email protected]
∗∗) E-mail: [email protected]
typeset using PTPTEX.cls hVer.0.9i
2 Letters Vol.***, No. *
and He burnings.26),2),3) However, there remained detailed comparison between ob-
servations and calculations related to successfulbursts, very low accretion rates, and
or superbursts (e.g., 1),23)).
Inthemean time, Fynboet al.16) (we call the referenceas Fynbo)revised the3α
rate of NACRE6) based on new experiments at high temperature of T > 109 K and
withanartificialextrapolationtotheverylowtemperatureregiontowardT ∼ 107 K.
However, in the present case of accreting neutron stars we can regard the differences
between NACRE and Fynbo as unimportant , because the OKK rate is much larger
for the temperature less than 108 K compared to the difference between NACRE
and Fynbo.
It is noted that although nuclear burning strongly depends on the temperature,
the density becomes very important at high densities of ρ ≥ 106 g cm−3 and low
temperatures of T ≤ 108 K due to the screening effects.11)
The new rate has been found to affect significantly the evolution of low-mass
and intermediate stars,7),8) where the evolutions from the zero-age main sequence
throughthecoreHeflash/burningfrom1M⊙ to10M⊙ havebeeninvestigated. From
the HR diagram obtained with the OKKrate, theresults disagrees in detail with the
observations. If the OKK rate in the temperature range of 108 K < T < 2×108 K
is exactly correct, we must invoke some new physical processes such as rotational
mixing and/or convection mechanism.
On the other hand, Saruwatari and Hashimoto17) have shown the important
difference between the ignition points calculated by the two rates for the helium
accreting carbon-oxygen whitedwarf. Theyhave concluded that forall theaccretion
rates, nuclear fuel ignites at the accumulated layers of helium and a scenario of Type
Ia supernovae changes for low accretion rates of M˙ < 4×10−8 M⊙ yr−1. Related to
this study, since the progenitor of Type Ia supernovae have been constructed with
useofpreviousreactionrate, discussionsconcerningtheoriginofthesupernovaethat
originate from the binary between a white dwarf and red giant could be changed.18)
Furthermore, simple helium ignition model on an accreting neutron star has
been investigated.15) It is concluded that the OKK rate does not fit the observations
of X-ray bursts due to the pure helium accretion but previous rate agrees with the
observations. However, their model is based on a model of one variable of column
density and only steady state has been followed until the ignition. It should be
noted that the whole structure from the center to the surface of a neutron star
must be solved to discuss the thermal evolution of accreting neutron stars. This has
been stressed by Fujimoto et al.19) Therefore, it is desirable to include the whole
structure and follow the evolution of accreting neutron stars with physical processes
included,19) if we try to elucidate the effects of the OKK rate on the evolution and
compare the results with the available observations. We note that for the combined
burning of H and He, no difference between Fynbo and OKK appears since the
burst is triggered at higher temperature region of T > 5×108 K.27) In this letter,
we present the results of the evolutionary calculation beyond the helium ignition
of a helium accreting neutron star. We will show the important role of both the
3-α reaction rates in the low temperature region and the properties of the crustal
heating.20),21)
May 2011 Letters 3
2. Evolution toward the explosive helium burning on the accreting neutron star
Accreting neutron stars are considered to be the origin of the type I X-ray
bursts.19) Accreting materials are usually hydrogen and/or helium. Since the hy-
drogen is converted to helium through steady hydrogen burning, helium is gradually
accumulated on the neutron star and the deep layers become hot and dense. Helium
flash triggered in the region composed of degenerate electrons could develop to the
dynamical stage, depending on the accretion rate M˙ .22) The properties of ignition
are determined from the thermal structure around the bottom of accreting layers. It
hasbeenfoundthatunstableheliumburningaftertheexhaustionofhydrogenresults
if the helium accretes in the range of 2×10−10M⊙ yr−1 < M˙ < 10−9M⊙ yr−1.23)
We have computed the evolution of an accreting neutron star by the Henyey-
type implicit-explicit method19) during the hydrostatic evolutionary stage of the
accretion of helium, where the full set of general relativistic equations of spherically
symmetric stars is based on the formulation by Thorne.24) Physical inputs are the
same as Ref 19) except for the network of nuclear burning and additional heating
from the crust. We note that under the assumption of the spherically symmetry, the
basic four equations of neutron star evolution becomes the same type of those of the
ordinary stellar evolution except for the effects of the general relativity.
Initial models are constructed by accreting helium on a neutron star of which
the gravitational mass and the radius of the neutron star are M = 1.3 M⊙ and R =
8.1 km, respectively. We can obtain the initial models for each constant accretion
rate shown in Table I. after the steady state has been achieved as in Ref. 19): the
integratedenergyfluxofthenon-homologouspartforthegravitationalenergyrelease
becomes negligible and therefore the thermal structure does not almost change. The
masses have been gradually increased during the accretion toward the steady state
and the amounts are less than 1 % of the total mass when the steady state has been
achieved.
Tofollowthenuclearburning,thealpha-networkhasbeenimplementedtoobtain
the nuclear energy generation rate:25) the network consists of 4He, 12C, 16O, 20Ne,
24Mg, 28Si, 32S, 40Ca, 44Ti, 48Cr, 52Fe, and 56Ni. We include and examine the
two reaction rates of Fynbo and OKK for the 3α reaction, and the other reaction
chain is selected from the faster reaction of either (α,γ) or (α,p) followed by the
(p,γ) reactions. After the calculations of stellar structure have been converged,
this network is operated to obtain the nuclear energy generation rate to calculate
the next stellar structure as the same method in Ref. 26). The nuclear energy
generation rate of the 3α reaction is crucial to determine the ignition condition and
would be proportional to T−3exp(−44.0/T )27) for the high temperature region of
9 9
T > 0.1. However, the temperature dependence cannot be written by a simple
9
analytical formula, that is,
3
Y
ε ∝ ρ2 fh3αi(T), (1)
3α
(cid:18)4(cid:19)
where Y is the helium mass fraction and f is the screening factor which is included
by the same method in Ref. 5). As a function of T, h3αi(T) was obtained from
the average with the Maxwell-Boltzmann distribution concerning the product of the
4 Letters Vol.***, No. *
10 10
Evolutionary track Evolutionary track
Ignition Ignition
9.5 Deflagration (Y=0.1) 9.5 Deflagration (Y=0.1)
Deflagration (Y=0.5) Deflagration (Y=0.5)
9 9
K) K)
(T 8.5 (T 8.5
g g
o o
l l
8 8
7.5 7.5
7 7
6 6.5 7 7.5 8 8.5 9 9.5 10 6 6.5 7 7.5 8 8.5 9 9.5 10
log r (g cm-3) log r (g cm-3)
Fig. 1. Evolutionary track of the temperature against the density at the bottom of the burning
layer in M˙ = 3×10−10M⊙yr−1 for the3α reaction rate of OKK(left panel) and Fynbo(right
panel). The ignition and deflagration curves are definedby εn =εrad and Eq.(8), respectively.
cross section and the relative velocity.4)
The importance of a crustal heating during the thermal evolution of neutron
stars has been pointed out.20) The heating rate is tabulated in Ref. 28) and the
heating rate is
q
Qi = 6.03M˙−10 i 1033 erg s−1, (2)
1MeV
where M˙−10 is accretion rate in units of 10−10 M⊙yr−1 and qi is deposited heat in
MeV/nucleon whose representative value within the present accretion rate in this
paper would be evaluated to be 0.012 MeV/nucleon from the produced average
heating rate:
< Q >= Q dm/∆ M, ∆M = dm, (3)
i i
Z Z
crust crust
where dm = 4πr2ρdr and the integral covers the crust of the density range 1011 −
1013 g cm−3 and ∆ M ≃ 1.2×10−4 M⊙ with the thickness of 1 km. We can easily
include it just as the neutrino energy loss rates and/or nuclear generation rates,
that is, the rate of the crustal heating is added in addition to the nuclear energy
generation rate, where we write the energy equation in Newtonian approximation
for simplicity:
∂L ∂s
r ∗
= ε −ε −T ,
∂M n ν (cid:18)∂t(cid:19)
r Mr
where L is the energy flow, ε∗ includes the nuclear energy generation rate, crustal
r n
heating rate, ε is the neutrino energy loss rate, s is the specific entropy.
ν
We have selected sixcases ofaccretion fortwo reaction rates (seeTableI),where
the hight of the accreted layer is at most some 10 m during the shell flash. We have
shown the evolutionary tracks in Fig. 1 for the OKK rate (left panel) and the Fynbo
rate (right panel) with the solid lines. The tracks correspond to the bottom of the
accretion layers, where the maximum temperature results and shell flash should be
triggered in the neighborhood of the region.
May 2011 Letters 5
Let us define the ignition and deflagration and/or detonation curves to show
clearly the points of the helium ignition and the development of the shell flash. The
density and the temperature in the below equations (4)-(8) must correspond to the
bottom of the burning layer and/or around that of the accreted layer. The energy
conservation law can be written as follows;
dT
c =ε −ε , (4)
P n rad
dt
wherec isthespecificheatattheconstantpressureandε isthenuclear generation
P n
rate of the 3α reaction. We approximate the radiative loss in the simple form,
4acT4
ε = , (5)
rad 3κ σ2
where κ is the opacity, σ is the column density, a is the radiation constant, and c
is the speed of light.29) The column density can be obtained from the pressure and
the gravitational acceleration for a neutron star model (see Eq. (9)). The ignition
curve is defined on the plane of (ρ,T) to satisfy the equality ε = ε . The ignition
n rad
curves are drawn in Fig. 1 for the OKK rate and the Fynbo rate, where we can infer
that helium ignition should be triggered at the much lower density.
The dynamical timescale τ is defined as
dyn
1
τ = , (6)
dyn
24πGρ
p
where G is the gravitational constant. On the other hand, the time scale τ of the
n
increase of the temperature due to the nuclear burning is defined to be
c T
P
τ = . (7)
n
ε
n
The deflagration curve has been obtained from the below condition:11)
τ = τ . (8)
n dyn
If τ < τ , deflagration wave should be originated, which could develop and burn
n dyn
out the previously accumulated layers (hereafter we call the wave the deflagration
one). The deflagration curves are also drawn on the plane of (ρ,T) in Fig. 1. We
note that the curves depend significantly on the helium mass fraction designated by
Y.
Figure 1 shows the evolutionary track on the plane of (ρ,T) toward the helium
ignitionforthebottomoftheburninglayer(aroundthebottomoftheaccretedlayer)
and subsequent helium flash beyond the deflagration curves, where the deflagration
curves of two cases of helium mass fractions (Y = 0.1 and 0.5) are drawn with use of
the OKK rate and Fynbo rate. We can find that the helium ignition occurs at lower
density regions by almost two orders of magnitude if the OKK rate is adopted. We
note that for M˙ < 3×10−8 M⊙ yr−1, the helium flash develops to the dynamical
6 Letters Vol.***, No. *
Table I. EnergyreleasesE peraburstforafixedaccretionratearegivenfortwocasesofOKK
burst
and Fynbo. The ignition pressure for each M˙, P , corresponds to the maximum temperature
ign
layer inside the accretion layers whose location is around thebottom of the accumulated ones.
reaction rate M˙ [M⊙ yr−1] logPign[dyn cm−2] Eburst[erg]
OKK 3×10−10 24.34 4.98×1040
8×10−10 23.07 2.72×1039
1×10−9 23.04 2.52×1039
5×10−9 23.03 2.47×1039
1×10−8 22.90 1.80×1039
3×10−8 22.68 1.09×1039
Fynbo 3×10−10 26.47 6.84×1042
8×10−10 24.35 5.13×1040
1×10−9 24.31 4.73×1040
5×10−9 23.83 1.51×1040
1×10−8 23.34 5.07×1039
3×10−8 22.68 1.09×1039
stage. Therefore, our hydrostatic evolution code cannot follow the dynamical stage
which would lead to the deflagration and/or detonation.
3. Comparison with the observations
Itwouldbeusefultojudgewhich3αrateismoreconsistentwithavailable obser-
vations. In general, it is very difficult to follow in the hydrodynamical calculations
theheliumflashuntilthefuelof heliumdepletes as describedintheprevioussection.
However, using the results of evolutionary calculations obtained in the previous sec-
tion, wecancomparequalitatively thetheoretical calculations withtheobservations.
We choose the layer of the helium ignition from which the pressure, the density, the
temperature, and the composition of helium are obtained. We define the density
ρ and the pressure P for the layer when the helium ignition occurs. We can
ign ign
get ρ from the crossing point between the evolutionary track of the bottom of the
ign
accreting layer and the ignition curve. As a consequence, P is obtained from the
ign
equation of state. Since the accumulated layers are well approximated by the plane
parallel configuration, we can adopt the approximation to estimate the accumulated
mass with a enough approximation; It is written in the spirit of the plane parallel
model as follows:30),31),32)
P = g σ , (9)
ign s ign
where g is the gravitational acceleration and σ is the the column density at the
s ign
ignition point. The energy release E can beobtained under the assumption that
burst
the accumulated layers on the ignition layer are completely burnt out:
E = ∆M Q /(1+z ), (10)
burst ign nuc s
= 4πR2σ Q /(1+z ). (11)
ign nuc s
May 2011 Letters 7
1043
OKK
Fynbo
1042
intermediate
g) long bursts
er
(urst1041
b
E
1040
4U 1820-30
1039
10-11 10-10 10-9 10-8 10-7
M · (Mo• yr-1)
Fig. 2. Energy release of E against the accretion rate M˙ for two cases of OKK and Fynbo.
burst
Observational values from different astrophysical objects which have been identified to cause
typeI X-raybursts dueto thepurehelium accretion, are plotted as was done in Ref. 15).
Here, ∆M is the total rest mass of the accreted layers, the gravitational redshift,
ign
z = (1−2GM /c2R)−1/2 −1 ≃ 0.38, and Q is the nuclear energy release per
s t nuc
nucleon (Q = 1.6 MeV/nucleon), where all fuel of helium are assumed to be
nuc
burnt into iron. For our model of the neutron star, since g = GM(1+z )/R2, we
s s
get logg =14.56.
s
In Table I, we show P and E in each accretion rate for the cases of OKK
ign burst
and Fynbo. In Fig. 2, we show E as a function of accretion rates M˙ , where are
burst
shown the two observations of type I X-ray bursts, 4U1820-30 (labeled by 4U1820-
30), SLX 1737-282, SLX 1735-269, and 2S 0918-549 (labeled by intermediate long
bursts). Their accreted matter from companions has been known to be pure he-
lium.33),34),35),36),37),38) As is written in Ref. 15), the uncertainties of accretion
rates and burst energies are indicated by boxes. For these three observations, we
can seefromFig. 2, resultsby OKKareconsistent withtheobservations if wechoose
the accretion rate of M˙ ≤ 3×10−10M⊙ yr−1.
4. Discussion and conclusions
The ignition densities that determine the triggering mechanism on helium ac-
creting neutron stars will be changed significantly if we adopt the new 3α reaction
rate for the low accretion rate of M˙ < 10−8M⊙ yr−1. Until the OKK rate has been
presented, the method by Nomoto11) has been used of which a simple extrapolation
of the Breit-Wigner type function to the low-temperature side is adopted, where it
has been advocated that the non-resonant 3α reaction is crucial in determining a
heliumignition ofcompactstarsforthelowaccretion rate. However, themicroscopic
calculation for the three body problem has been found to be crucial in evaluating
the 3α reaction rate. In the present paper, we have examined the effects of the OKK
rate on the very low temperature site of astrophysics. Although our results seem to
contradict with those of Peng and Ott,15) we may ascribe it to the following. First,
we have adopted the rate of the crustal heating from Ref. 28). Second, they have
used the mass of 1.4 M⊙ and the radius of 10 km for a neutron star. Third, while
8 Letters Vol.***, No. *
1)1023
-s
-2cm 1022 i n ltoenrgm beudriastte 4U 1820-30
g
st (er1021
u
m cr1020
o
ux fr1019
at fl Our models
He1018
10-11 10-10 10-9 10-8 10-7
M · (Mo• yr-1)
Fig. 3. Energy flux in units of erg cm−2s−1 from the crust against accretion rates with use of the
table28) (solid line) and the two boxesare taken from Ref. 15).
we have followed the evolution of an accreting neutron star from the center to the
surface of the star, they have solved only steady-state equations above the crust as
a function of a single dependent variable of the column density. As a consequence,
it has been written that the outward energy release due to the crustal heating was
taken to be a free parameter as a boundarycondition, which would be inappropriate
in the calculations of accreting neutron stars.
The most important difference comes from the treatment of the crustal heat-
ing. In the present case, the energy flux (erg cm−2s−1) from the crust is shown in
Fig. 3. The average energy per unit mass from the crust can be evaluated to be
∼ 0.01 MeV/nucleon as shown in the previous section. On the contrary, the rate
they have adopted could be nearly 0.1 MeV/nucleon.21) These differences reflect the
calculations of the heat flux as shown in Fig. 3, where the solid line represents heat
fluxinourcalculations andthetwoboxesaretaken fromRef.15). Itisclear thatthe
heat flux from the crust in the present study is 1-2 orders of magnitude lower than
the case in Ref 15). As a consequence, our initial models just before the accretion
starts might have lower temperatures compared to those used in Ref. 15). The most
significant difference would come from the initial model and therefore, the location
of the ignition point on the (ρ,T) plane. While the bottom of the accreted layers in
our initial model for M˙ = 3×10−10M⊙ yr−1 has the temperature of T = 1.6×107 K
(see Fig. 1.), we can infer for the case of Ref. 15) the temperature could be rather
high. From the column density obtained from the box of the observation of interme-
diate long bursts (see Fig. 2 in their paper), we can get the ignition temperature of
T = (1−2)×108 K for the Fynborate. If the initial temperatureof their calculation
take nearly the same temperature, which is the common feature of the accreting
compact objects before the beginning of the shell flashes,39) ignition density is found
to be in the neighborhood of that for the OKK rate. Contrary to our calculations
which are the results of stellar evolution and include all the necessary physical pro-
cesses, their calculations depend essentially on the numerically convenient boundary
condition at the assumed surface of the crust.
May 2011 Letters 9
We have shown that OKK rate must change the scenario of Type Ia supernovae
and type I X-ray bursts for the low accretion rates, where the temperature at the
ignitionislessthan108 K.Sincetheratewascalculatedunderthesituationofthreeα
particlesinvacuum,applicationoftheOKKrateisdoubtfulagainstthehighdensity.
As seen from Fig. 3 in Ref. 4), the discretized continuum wave functions of the two
α particle system extend to 5×103 fm. Therefore, in the medium of stellar plasma,
effects more than three α particles cannot be negligible for ρ > 106 g cm−3.40) For
example, the reaction rate is to be calculated with the inclusion of an appropriated
screening potential. If the effects of many body interactions are included, the OKK
rate could be decreased by orders of magnitude. We may constrain the 3α reaction
rate from the astrophysical observations of type I X-ray bursts.
As example of constraint the OKK rate, if we adopt the upper values of the
observational box of Eburst inFig. 2for M˙ = 3×10−10M⊙ yr−1, we obtain logPign ∼
24.9 from (9)–(11). Since the evolutionary paths are almost the same for the two 3α
rates until the ignition points (see Figs. 1), the corresponding ignition density with
the evolutionary paths of OKK becomes in the range of logρ = 7.7−7.8 for the
ign
OKK rate reduced by a factor of 102−3. We note that the properties of shell flash
depend to some extent on the structure of a neutron star (mass and radius); Since
Eburst is proportional to R4/M, a rather hard EOS (R = 12 km and M = 2.0 M⊙)
givesthreetimeslargervaluecomparedtoourcase,whichwouldbepreferableforthe
OKK rate. On the other hand, it depends on the non-standard cooling processes:
if we use the non-standard cooling processes such as a pion-condensation,19) our
estimates would be changed significantly.
In conclusion, we have found that the new rate affects the helium ignition in ac-
cretingneutronstars,whereforlower accretion rates, heliumburnsatlower densities
and temperatures, which should change the epoch of the formation of a helium de-
flagration wave and the modeling of type I X-ray bursts.5) In particular, we consider
thatthecrustalheatingaffects theignitionpropertiessignificantly. Furthermore,the
mechanism behind superbursts should be studied again using the new rate, because
the amount of 12C could play a crucial role in inducing the superbursts.41)
Acknowledgements
This work has been supported in part by a Grant-in-Aid for Scientific Research
(19104006, 21540272) of the Ministry of Education, Culture, Sports, Science and
Technology of Japan.
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